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Seeking help with f(x) question. Thanks!

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Seeking help with f(x) question. Thanks!

by Orla M » Sat Feb 16, 2013 5:44 am
Question asks:

For which of the following functions of f is f(x) = f(1-x) for all values of x?

a) f(x)=1-x
b) f(x) = 1-x2 (x squared)
c) f(x) x2 - (1-x)2 (in both instances the value of 2 represents the number squared)
d) f(x) = x2 (1-x)2 (again 2 means squared here)
e) f(x) = x/1-x

The answer is D. I started to solve using two values for x but that didn't lead me to the right answer. How can I approach this question to solve?

Many thanks
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by GMATGuruNY » Sat Feb 16, 2013 7:31 am
For which of the following functions f is f(x) = f(1-x) for all x?

a. f(x)= 1-x
b. f(x)= 1-x²
c. f(x)= x²-(1-x)²
d. f(x)= x²(1-x)²
e. f(x)= x/(1-x)
Let x=2.
Then f(x) = f(2) and f(1-x) = f(1-2) = f(-1).
The question becomes:

For which of the following functions does f(2) = f(-1)?

Answer choice A:
f(2) = 1-2 = -1.
f(-1) = 1-(-1) = 2.
Doesn't work.

Answer choice B:
f(2) = 1 - 2² = -3.
f(-1) = 1 - (-1)² = 0.
Doesn't work.

Answer choice C:
f(2) = 2² - (1-2)² = 4 - 1 = 3.
f(-1) = (-1)² - [1-(-1)]² = 1-4 = -3.
Doesn't work.

Answer choice D:
f(2) = 2² * (1-2)² = 4 * 1 = 4.
f(-1) = (-1)² * [1-(-1)]² = 1 * 4 = 4.
Success!

Answer choice E:
f(2) = 2/(1-2) = -2.
f(-1) = (-1)/[(1-(-1)] = -1/2.
Doesn't work.

The correct answer is D.
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by Orla M » Sat Feb 16, 2013 7:40 am
Excellent, thanks Mitch. Now the light bulb has switched on - I was missing that second step in each testing of the answers.
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by hemant_rajput » Sat Feb 16, 2013 9:01 am
Orla M wrote:Question asks:

For which of the following functions of f is f(x) = f(1-x) for all values of x?

a) f(x)=1-x
b) f(x) = 1-x2 (x squared)
c) f(x) x2 - (1-x)2 (in both instances the value of 2 represents the number squared)
d) f(x) = x2 (1-x)2 (again 2 means squared here)
e) f(x) = x/1-x

The answer is D. I started to solve using two values for x but that didn't lead me to the right answer. How can I approach this question to solve?

Many thanks

I've one more approach.

a.

f(x)= 1-x
say y= 1-x, just to make it more clear.
f(y) = 1 -y
substituting value of y
f(1-x) = 1 - (1-x)
f(1-x) = 1 - 1 + x
f(1-x) = x
f(x) not equal to f(x-1)

b.
f(x) = 1-x^2
f(1-x) = 1 - (1-x)^2 => 1 - (1 + x^2 - 2x)=> 2x - x^2

f(x) not equal to f(x-1)

c.
f(x) = x^2 - (1-x)^2
f(1-x) = (1-x)^2 - (1-(1-x))^2 => (1-x)^2 - (1- 1 + x))^2 => (1-x)^2 - (x)^2

f(x) not equal to f(x-1)

d.
f(x) = x^2 * (1-x)^2
f(1-x) = (1-x)^2 * (1-(1-x))^2 => (1-x)^2 * (1-1+x)^2=>(1-x)^2 * x^2

bingo

f(x) = f(1-x)
I'm no expert, just trying to work on my skills. If I've made any mistakes please bear with me.
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by pemdas » Sat Feb 16, 2013 9:21 am
one function becomes another function, this is meant by the expression. f(x)=not (1-x) but f(1-x). We are looking for the values of all x to satisfy two functions not defined precisely - only their arguments are given (as x and x-1). By supplying x and x-1 into the functions below we should get the same answers and select the right choice. However, since we don't want to be bogged into calculations we simply set x=0 (x can be any value)
a) f(x)=1-x ==> f(1-x)=1-(1-x)=-x but not 1-x Wrong (even without supplying x=0 here)
b) f(x) = 1-x^2 ==> f(1-x)=1-(1-x)^2=1-(1-2x+x^2)=2x-x^2 but not 1-x^2 Wrong
c) f(x) = x^2-(1-x)^2 ==> f(1-x)=(1-x)^2-(1-(1-x))^2= 1-2x+x^2-(1-(1-2x+x^2)= -4x+2x^2 but not x^2-(1-x)^2 Wrong
d) f(x) = (x^2)*(1-x)^2 ==> f(1-x)=(1-x)^2 *(1-(1-x))^2
we supply x=0 here to make it faster, f(0)=0^2*(1-0)^2=0 and f(0-1)=(0-1)^2 *(1-(1-0))^2=0 good choice
e) f(x) = x/1-x
here too supply x=0 and get f(0)=0 and f(0-1)=(0-1)/(1-(0-1))=-1/2 but not 0 Wrong

answer d
Orla M wrote:Question asks:

For which of the following functions of f is f(x) = f(1-x) for all values of x?

a) f(x)=1-x
b) f(x) = 1-x2 (x squared)
c) f(x) x2 - (1-x)2 (in both instances the value of 2 represents the number squared)
d) f(x) = x2 (1-x)2 (again 2 means squared here)
e) f(x) = x/1-x

The answer is D. I started to solve using two values for x but that didn't lead me to the right answer. How can I approach this question to solve?

Many thanks
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